CONGRUENCE 115

the least common multiple is a common multiple of the two
integers.
For the direct theorem we have n > |ab| : (¢ X b)

§§ 269, 70.
And labl : (a GOB) =a X-b § 271.
Hence n>3>aXb
For the converse n > a X b > aDn>>a § 65.

273. Th. The common multiples of two integers are the same
as the multiples of their least common multiple.
This follows immediately from the preceding theorem.

274. Th. a-=0~b-=0>D (a®b)(a X b) = |abd|

275. Th. a-=0~b-=0~allbDa X b= |ab|
For example, 5§ X 6 = 5 X 6 = 30

276. Th. a=b(modm)~a=b(modn)Da=0b(mod mXn);
and conversely.

For(e —b0>m~—~a—-0>3n)=@—0>mXmn). §272.

Ex. 1. Verify that 7 =1 (mod 2), 7 = 1 (mod 3)
and 7 =1 (mod 2 X 3)
Ex. 2. Verify that 20 is congruent to 2 with respect to the
three moduli 6, 9 and 6 X 9.
Ex. 3. Verify that 1128 is congruent to — 2000 with re-
spect to the moduli 68, 92, and 68 X 92.

277. Th. [e = b (mod m;) —~ a = b (mod m-)
~a = b (mod ms) —~ -]
= [a = b (mod my X mg X m3 X +++)]
For any common multiple of any number of integers is a
multiple of their least common multiple, and conversely (1).

278. Th. a=b(modm) ~a > c~b>cDa=b(mod mX c)
Fora>c~b>cDa= b (modc)
Hencea =b(modm) ~a>>c~b>3>cD

a=>b(modm) ~a=b(modc) Da=>b(modm X c)
Example. 20=8 (mod 6),20>>4,8>> 4, 20=8 (mod 6 X 4)

(1) § 718, Theory of Integers.